Survival Analysis via Hazard Based Modeling and Generalized Linear Models

The connection between survival analysis via hazard based modelling and generalized linear models had been made very early even since the description of the proportional hazard (PHM) Cox (1972) and generalized linear models (GLM) Nelder and Wedderburn (1972). For example,

– Breslow (1974) considers the proportional hazard model as a discrete time logistic regression in which discrete probability masses are put on the (ordered) set of observed failure times

– Breslow (1972) and Breslow (1974) discretize a continuous time model by forcing the hazard to be constant between successive failure times. This piecewise exponential model (PEM) was later recognized to yield estimates identical to Poisson regression for rate/exposure data Laird and Olivier (1981).

– Holford (1976) and Holford (1980) introduce Poisson regression models (PRM) for survival analysis and prove the equality of Maximum Likelihood Estimates (MLE) from the PH, PR and the PE models.

– the logistic regression model has also been applied to model survival probabilities very early after the introduction of GLM and PHMs Brown (1975); Mantel and Hankey (1978); Thompson (1977).

– It appears that the connection between survival analysis, lifetable methods and GLMs (Poisson and Logistic) has been utilized in a number of early publications to fit the proportional model Aitkin and Clayton (1980); Peduzzi et al. (1979); Whitehead (1980) . At least one early landmark clinical trial has been analyzed with the PHM using software developed for GLMsLowrie et al. (1981).

– a thorough and rather general treatment of the relationship between survival analysis, the PHM, logistic and Poisson regression was given in Efron (1988) who also reviewed the relevant literature up to that time. In this latter paper, commonalities among the GLM approach to survival analysis were highlighted: the explicit discretization (or partition) of the time axis to disjoint intervals, the introduction of specific parameterization of the underlying hazard function (usually via step functions)

– A survey of the more recent literature appears in Section 3.19 of Ruppert et al. (2009)

Literature

  • Cox, D. R., “Regression Models and Life-Tables”, Journal of the Royal Statistical Society. Series B (Methodological) (1972), 187–220.
  • J. A. Nelder and R. W. M. Wedderburn, “Generalized Linear Models”, Journal of the Royal Statistical Society. Series A (General) (1972), 370–384.
  • N Breslow, “Contribution to the discussion on the paper of D.R. Cox : “Regression Models and Life-Tables””, Journal of the Royal Statistical Society. Series B (Methodological) (1972), 216–217.
  • N Breslow, “Covariance analysis of censored survival data”, Biometrics (1974), 89–99.
  • T R Holford, “Life tables with concomitant information”, Biometrics (1976), 587–597.
  • Theodore R. Holford, “The Analysis of Rates and of Survivorship Using Log-Linear Models”, Biometrics (1980), 299–305.
  • Charles C. Brown, “On the Use of Indicator Variables for Studying the Time-Dependence of Parameters in a Response-Time Model”, Biometrics (1975), 863–872.
  • Nathan Mantel and Benjamin F. Hankey, “A logistic regression analysis of response-time data where the hazard function is time dependent”, Communications in Statistics – Theory and Methods (1978), 333.
  • W A Thompson, “On the treatment of grouped observations in life studies”, Biometrics (1977), 463–470.
  • Murray Aitkin and David Clayton, “The Fitting of Exponential, Weibull and Extreme Value Distributions to Complex Censored Survival Data Using GLIM”, Journal of the Royal Statistical Society. Series C (Applied Statistics) (1980), 156–163.
  • P Peduzzi and T Holford and R Hardy, “A computer program for life table regression analysis with time dependent covariates”, Computer Programs in Biomedicine (1979), 106–114.
  • John Whitehead, “Fitting Cox’s Regression Model to Survival Data using GLIM”, Journal of the Royal Statistical Society. Series C (Applied Statistics) (1980), 268–275.
  • E G Lowrie and N M Laird and T F Parker and J A Sargent, “Effect of the hemodialysis prescription of patient morbidity: report from the National Cooperative Dialysis Study”, The New England Journal of Medicine (1981), 1176–1181.
  • Bradley Efron, “Logistic Regression, Survival Analysis, and the Kaplan-Meier Curve”, Journal of the American Statistical Association (1988), 414–425.
  • David Ruppert and M.P. Wand and Raymond J. Carroll, “Semiparametric regression during 2003 – 2007”, Electronic journal of statistics (2009), 1193–1256.
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